Real Time Determination of Casing Location and Distance with Tilted Antenna Measurement

ABSTRACT

Methods and apparatus for detecting nearby conductors such as pipes, well casing, etc., from within a borehole. A nearby casing string can be detected by transmitting an electromagnetic signal from a first antenna on a downhole logging tool and measuring a response signal with a second antenna. As the tool rotates, the transmitting and measuring are repeated to determine the azimuthal dependence of the response signal. The azimuthal dependence is analyzed to determine an diagonal component and a cross component. The amplitude of the diagonal component is indicative of distance to the conductive feature. Direction can be determined based on the diagonal component alone or in combination with the cross component. Sinusoidal curve fitting can be employed to improve accuracy of the distance and direction estimates. At least one of the antennas is preferably tilted. Measurement results are presented for parallel tilted and perpendicular tilted antennas.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority to Provisional U.S. Application61/357,324, titled “Real time determination of casing location anddistance with tilted antenna measurement” and filed Jun. 22, 2010 by M.Bittar, S. Li, and H. Wu, which is hereby incorporated herein byreference.

BACKGROUND

The world depends on hydrocarbons to solve many of its energy needs.Consequently, oil field operators strive to produce and sellhydrocarbons as efficiently as possible. Much of the easily obtainableoil has already been produced, so new techniques are being developed toextract less accessible hydrocarbons. These techniques often involvedrilling a borehole in close proximity to one or more existing wells.One such technique is steam-assisted gravity drainage (“SAGD”) asdescribed in U.S. Pat. No. 6,257,334, “Steam-Assisted Gravity DrainageHeavy Oil Recovery Process”. SAGD uses a pair of vertically-spaced,horizontal wells less than 10 meters apart, and careful control of thespacing is important to the technique's effectiveness. Other examples ofdirected drilling near an existing well include intersection for blowoutcontrol, multiple wells drilled from an offshore platform, and closelyspaced wells for geothermal energy recovery.

One way to direct a borehole in close proximity to an existing well is“active ranging” in which an electromagnetic source is located in theexisting well and monitored via sensors on the drillstring. By contrastsystems that locate both the source and the sensors on the drillstringare often termed “passive ranging”. Passive ranging may be preferred toactive ranging because it does not require that operations on theexisting well be interrupted. Existing passive ranging techniques relyon magnetic “hot spots” in the casing of the existing well, which limitsthe use of these techniques to identify areas where there is asignificant and abrupt change in the diameter of casing or where thecasing has taken on an anomalous magnetic moment, either bypre-polarization of the casing before it is inserted into the wellbore,or as a random event. See, e.g., U.S. Pat. No. 5,541,517 “A method fordrilling a borehole from one cased borehole to another cased borehole”.In order to carry out such a polarization without interruptingproduction, it has been regarded as necessary to polarize the casing atsome point in the construction of the well. This approach cannot beapplied to wells that are already in commercial service withoutinterrupting that service.

BRIEF DESCRIPTION OF THE DRAWINGS

A better understanding of the various disclosed embodiments can beobtained when the following detailed description is considered inconjunction with the accompanying drawings, in which:

FIG. 1 shows an illustrative drilling environment in whichelectromagnetically-guided drilling may be employed;

FIG. 2 shows an illustrative ranging tool embodiment;

FIGS. 3A-3B illustrates the variables used in analyzing the operation ofthe tool;

FIG. 4 shows a coordinate system for specifying a direction and distanceto a nearby casing;

FIG. 5 is a flow diagram of an illustrative ranging method;

FIG. 6 shows an alternative ranging tool embodiment;

FIGS. 7A-7D are graphs of measured response signals;

FIGS. 8A-8D are graphs of a measured diagonal component;

FIGS. 9A-9D are graphs of a measured cross component FIGS. 10A-10B aregraphs of a first modeled distance dependence; and

FIGS. 11A-11B are graphs of a second modeled distance dependence.

While the invention is susceptible to various modifications andalternative forms, specific embodiments thereof are shown by way ofexample in the drawings and will herein be described in detail. Itshould be understood, however, that the drawings and detaileddescription are not intended to limit the disclosure to these particularembodiments, but on the contrary, the intention is to cover allmodifications, equivalents and alternatives falling within the scope ofthe appended claims.

DETAILED DESCRIPTION

The issues identified in the background are at least partly addressed bydisclosed methods and apparatus for detecting nearby conductors such aspipes, well casing, etc., from within a borehole. In at least somemethod embodiments, a conductive feature can be detected by transmittingan electromagnetic signal from a first antenna on a downhole loggingtool and measuring a response signal with a second antenna on thedownhole logging tool. As the tool rotates, the transmitting andmeasuring are repeated to determine the azimuthal dependence of theresponse signal. The azimuthal dependence is analyzed to determine andiagonal component and optionally a cross component. The amplitude ofthe diagonal component is indicative of distance to the conductivefeature. Direction can be determined based on the diagonal componentalone or, to eliminate any ambiguity, based on the diagonal component incombination with the cross component. Sinusoidal curve fitting can beemployed to improve accuracy of the distance and direction estimates. Atleast one of the antennas is preferably tilted. Measurement results arepresented for parallel tilted and perpendicular tilted antennas.

The disclosed systems and methods are best understood in a suitableusage context. Accordingly, FIG. 1 shows an illustrative geosteeringenvironment. A drilling platform 2 supports a derrick 4 having atraveling block 6 for raising and lowering a drill string 8. A top drive10 supports and rotates the drill string 8 as it is lowered through thewellhead 12. A drill bit 14 is driven by a downhole motor and/orrotation of the drill string 8. As bit 14 rotates, it creates a borehole16 that passes through various formations. A pump 20 circulates drillingfluid through a feed pipe 22 to top drive 10, downhole through theinterior of drill string 8, through orifices in drill bit 14, back tothe surface via the annulus around drill string 8, and into a retentionpit 24. The drilling fluid transports cuttings from the borehole intothe pit 24 and aids in maintaining the borehole integrity.

The drill bit 14 is just one piece of a bottom-hole assembly thatincludes one or more drill collars (thick-walled steel pipe) to provideweight and rigidity to aid the drilling process. Some of these drillcollars include logging instruments to gather measurements of variousdrilling parameters such as position, orientation, weight-on-bit,borehole diameter, etc. The tool orientation may be specified in termsof a tool face angle (a.k.a. rotational or azimuthal orientation), aninclination angle (the slope), and a compass direction, each of whichcan be derived from measurements by magnetometers, inclinometers, and/oraccelerometers, though other sensor types such as gyroscopes mayalternatively be used. In one specific embodiment, the tool includes a3-axis fluxgate magnetometer and a 3-axis accelerometer. As is known inthe art, the combination of those two sensor systems enables themeasurement of the tool face angle, inclination angle, and compassdirection. In some embodiments, the tool face and hole inclinationangles are calculated from the accelerometer sensor output. Themagnetometer sensor outputs are used to calculate the compass direction.

The bottom-hole assembly further includes a ranging tool 26 to induce acurrent in nearby conductors such as pipes, casing strings, andconductive formations and to collect measurements of the resulting fieldto determine distance and direction. Using these measurements incombination with the tool orientation measurements, the driller can, forexample, steer the drill bit 14 along a desired path 18 relative to theexisting well 19 in formation 46 using any one of various suitabledirectional drilling systems, including steering vanes, a “bent sub”,and a rotary steerable system. For precision steering, the steeringvanes may be the most desirable steering mechanism. The steeringmechanism can be alternatively controlled downhole, with a downholecontroller programmed to follow the existing borehole 19 at apredetermined distance 48 and position (e.g., directly above or belowthe existing borehole).

A telemetry sub 28 coupled to the downhole tools (including ranging tool26) can transmit telemetry data to the surface via mud pulse telemetry.A transmitter in the telemetry sub 28 modulates a resistance to drillingfluid flow to generate pressure pulses that propagate along the fluidstream at the speed of sound to the surface. One or more pressuretransducers 30, 32 convert the pressure signal into electrical signal(s)for a signal digitizer 34. Note that other forms of telemetry exist andmay be used to communicate signals from downhole to the digitizer. Suchtelemetry may employ acoustic telemetry, electromagnetic telemetry, ortelemetry via wired drillpipe.

The digitizer 34 supplies a digital form of the telemetry signals via acommunications link 36 to a computer 38 or some other form of a dataprocessing device. Computer 38 operates in accordance with software(which may be stored on information storage media 40) and user input viaan input device 42 to process and decode the received signals. Theresulting telemetry data may be further analyzed and processed bycomputer 38 to generate a display of useful information on a computermonitor 44 or some other form of a display device. For example, adriller could employ this system to obtain and monitor drillingparameters, formation properties, and the path of the borehole relativeto the existing borehole 19 and any detected formation boundaries. Adownlink channel can then be used to transmit steering commands from thesurface to the bottom-hole assembly.

FIG. 2 shows an illustrative ranging tool 202 in more detail. Itincludes a transmit antenna coil 204 set in a recess 206 around thecircumference of the tool. The illustrated transmit antenna 204 ispositioned at a 45° tilt angle to provide an azimuthal asymmetry to thetransmitted electromagnetic signal. The tool further includes two tiltedreceive antenna coils 210, 212 in a second recess around the toolcircumference. Antenna 212 is parallel to the transmit antenna 204,while antenna 210 is perpendicular to the transmit antenna. Antennas 210and 212 are shown as being collocated, but this is not a requirement.The disclosed methods can be employed with a single transmit-receiveantenna pair, which can be collocated if desired, but it is expectedthat the use of additional transmit-receive antenna pairings willprovide better ranging performance. As will become clear, the relativespacings and relative tilt angles can be varied as desired, so long asat least one of the transmit or receive antennas provides azimuthalsensitivity. A nonconductive filler material may be used to fill therecesses to seal and protect the antenna coils.

FIG. 3A shows an illustrative tool model having a longitudinal axiscoincident with a coordinate z-axis. A transmit antenna coil T1 isprovided with a tilt angle θ_(T1) relative to the z-axis and a receiveantenna coil R is provided with a tilt angle θ_(R) relative to thez-axis, usually with its normal vector in the same plane defined by thez-axis and the normal vector of the transmit antenna coil. The transmitand receive antenna coils are centered on the z-axis with their centerpoints separated by a distance d. The x- and y-axes are as shown in FIG.3B. The x-axis is directed from the z-axis toward the high side of theborehole. (For vertical boreholes, the north side of the borehole isoften taken as the “high” side.) The y-axis is drawn perpendicular tothe x- and z-axes using the right hand rule. The azimuthal angle β ismeasured from the x-axis starting in the direction of the y-axis. Themeasurements taken around the circumference of the borehole are oftengrouped into azimuthal bins. As illustrated in FIG. 3B, each bin i canbe associated with a representative azimuthal angle β_(i). Of course themeasurements can be grouped into bins along the z-axis as well.

The following equations use the notation V_(R) ^(T) to represent thesignal measured by a receive antenna coil R in response to the operationof a transmit antenna coil T. Where T is x, y, or z, V_(R) ^(T) assumesa hypothetical transmit antenna coil oriented along the x-, y-, orz-axis, respectively. The same is true where R is x, y, or z. Where thenormal vectors of the transmit and receive antenna coils are in the sameplane, the receive signal as a function of azimuthal angle β is:

$\begin{matrix}{{{V_{R}^{T}(\beta)} = {{\begin{bmatrix}{\sin \; \theta_{t}\cos \; \beta} \\{\sin \; \theta_{t}\sin \; \beta} \\{\cos \; \theta_{t}}\end{bmatrix}^{T}\begin{bmatrix}V_{x}^{x} & V_{y}^{x} & V_{z}^{x} \\V_{x}^{y} & V_{y}^{y} & V_{z}^{y} \\V_{x\;}^{z} & V_{y}^{z} & V_{z}^{z}\end{bmatrix}}\begin{bmatrix}{\sin \; \theta_{r}\cos \; \beta} \\{\sin \; \theta_{r}\sin \; \beta} \\{\cos \; \theta_{r}}\end{bmatrix}}},} & (1)\end{matrix}$

where the matrix elements V_(I) ^(J) are complex values representing thesignal amplitude and phase shift measured by a hypothetical receiverhaving a I-axis dipole component in response to the firing of ahypothetical transmitter having a J-axis dipole component.

Equation (1) can be also written out to highlight the azimuthal angledependence:

V_(R) ^(T)(β)=a_(xx) cos² β+(a_(xy)+a_(yx))cos β sinβ+(a_(xz)+a_(zx))cos β+a_(yy) sin²β+(a_(yz)+a_(zy))sin β+a_(zz)  (2)

where

a_(xx)=V_(x) ^(x) sin θ_(t) sin θ_(r); a_(xy)=V_(y) ^(x) sin θ_(t) sinθ_(r); a_(xz)=V_(z) ^(x) sin θ_(t) cos θ_(r)

a_(yx)=V_(x) ^(y) sin θ_(t) sin θ_(r); a_(yy)=V_(y) ^(y) sin θ_(t) sinθ_(r); a_(yz)=V_(z) ^(y) sin θ_(t) cos θ_(r).

a_(zx)=V_(x) ^(z) cos θ_(t) sin θ_(r); a_(zy)=V_(y) ^(z) cos θ_(t) sinθ_(r); a_(zz)=V_(z) ^(z) cos θ_(t) cos θ_(r)

Note that the a_(IJ) coefficients are determined by the antenna systemdesign and environmental effects, and they do not vary with azimuthalangle. Further manipulation yields:

$\begin{matrix}{{V_{R}^{T}(\beta)} = {{( {\frac{a_{xx}}{2} - \frac{a_{yy}}{2}} )\cos \; 2\beta} + {( {\frac{a_{xy}}{2} + \frac{a_{yx}}{2}} )\sin \; 2\beta} + {( {a_{xz} + a_{zx}} )\cos \; \beta} + {( {a_{yz} + a_{zy}} )\sin \; \beta} + ( {a_{zz} + \frac{a_{xx}}{2} + \frac{a_{yy}}{2}} )}} & (3)\end{matrix}$

Typical logging applications employ the azimuthal angle binningdescribed previously, which would cause each occurrence of the azimuthalangle β in equation (3) to be replaced with the representative azimuthalbin angle β_(i).

The left side of FIG. 4 shows a default x-y-z coordinate system for tool402, with an existing well casing 404 lying parallel to the z-axis at adistance L and an azimuthal angle φ. Equation (3) assumes an arbitrarycoordinate system and consequently would apply. However, if the defaultcoordinate system is rotated as shown in the right side of FIG. 4, i.e.,so that the azimuthal angle to the casing φ=0, the a_(xy), and a_(yz),and a_(zy) in the rotated coordinate system would be expected todisappear. Consequently the measured voltages would be expected to havea simplified representation:

$\begin{matrix}{{V_{R}^{T}( \beta_{i}^{\prime} )} = {{( {\frac{a_{xx}^{\prime}}{2} - \frac{a_{yy}^{\prime}}{2}} )\cos \; 2\beta_{i}^{\prime}} + {( {a_{xz}^{\prime} + a_{{zx}\;}^{\prime}} )\cos \; \beta_{i}^{\prime}} + ( {a_{zz}^{\prime} + \frac{a_{xx}^{\prime}}{2} + \frac{a_{yy}^{\prime}}{2}} )}} & (4)\end{matrix}$

where α′_(I,J) is tool coefficient and β′ is tool azimuthal angle in therotated coordinate system.

To achieve the simplification given in equation (4), a curve fittingoperation may be employed to determine an appropriate coordinaterotating angle φ₁, which also corresponds to the azimuthal angle for thedirection vector to the casing. This observation motivates the rangingmethods represented by the flowchart in FIG. 5.

Beginning in block 502, the tool begins its measurement cycle byselecting a first transmit antenna. In block 504, the tool transmits anelectromagnetic signal with the selected transmit antenna and measuresthe response of each receive antenna. The tool also determines itsposition and orientation at the time of transmission. In block 506, thetool updates the measurement averages for the bin corresponding to thattool position and orientation. In block 508, the tool determines whethera measurement cycle has been completed (i.e., whether each of thetransmit antennas has been used), and if not, blocks 504-508 arerepeated until the measurement cycle is finished.

In block 510, the azimuthal dependence of the measurements is analyzedto find three components: the diagonal component Vdiag, thecross-component Vcc, and the constant component Vconst. These componentsare defined as:

$\begin{matrix}{{V_{diag}(i)} \equiv \frac{{V_{R}^{T}( \beta_{i} )} + {V_{R}^{T}( \beta_{i \pm \frac{N}{2}} )}}{2}} & (5) \\{{V_{cc}(i)} \equiv \frac{{V_{R}^{T}( \beta_{i} )} - {V_{R}^{T}( \beta_{i \pm \frac{N}{2}} )}}{2}} & (6) \\{V_{const} \equiv \frac{\sum\limits_{i = 1}^{N}{V_{R}^{T}( \beta_{i} )}}{N}} & (7)\end{matrix}$

where N is the number of azimuthal bins (FIG. 3B) and bin i±N/2 is thebin opposite from bin i. Equation (7) corresponds to the third term inequation (4), equation (6) corresponds to the second term in equation(4), and equation (5) corresponds to the sum of the first and thirdterms in equation (4). Equations (5)-(7) do not account for the rotationangle φ, but the system determines that angle in block 512 by, e.g.,fitting sinusoidal curves to the diagonal component and cross-component.The curve fits can be performed to each component separately or, ifdesired, to the measurements directly. The curve fit yields parametersA, B, C, and φ:

V _(R) ^(T)(β)=A cos 2(β−φ)+B cos(β−φ)+C  (8)

A least mean square curve fitting method was employed, but other fittingtechniques may also be suitable.

In a homogeneous medium, the above three components are only sensitiveto a nearby casing, especially the diagonal component Vdiag. Thediagonal component is expected to demonstrate better sensitivity to anearby casing and better noise resistance, but due to its double period(cos 2(β−φ)) will also have a 180° ambiguity for measuring the casing'sazimuthal angle φ. Since the cross-component Vcc only has a singleperiod (cos(β−φ)), it can be used to resolve this ambiguity for a uniquedetermination of the azimuthal angle φ. The distance can then beestimated based on the amplitudes of the components. An example of thisdirection and determination process is described further below.

Once the system has determined a measurement of casing direction anddistance in block 512, the new measurement can be used in block 514 toupdate a display for the driller and/or to automatically adjust thesteering direction for the drilling assembly. In block 516, the tooldetermines whether operations are complete, and if not, repeats theprocess.

FIG. 6 shows a ranging tool embodiment that was tested in a water tankhaving water with a resistivity of 1 Ω·m. The tested tool included twotransmit-receive antenna pairs, the first pair being parallel (Tup-Rx inFIG. 6) with both tilted at an angle of −45° and the second pair beingperpendicular (Tdn-Rx in FIG. 6) with the transmit antenna coil tiltedat an angle of 45° and the receive antenna coil tilted at an angle of−45°. The spacing between the first transmit-receive antenna pair is d1and the spacing between the second transmit-receive antenna pair is d2.Measurements were made with d1=d2=48″ and a signal frequency of 125 kHz.The casing was placed parallel to the tool as shown in FIG. 4.

The measured response signals are graphed in FIGS. 7A-7D. FIGS. 7A and7B show the real and imaginary parts of the response signal for theparallel transmit-receive antenna pair, while FIGS. 7C and 7D show thereal and imaginary parts of the response signal for the perpendiculartransmit-receive antenna pair. In each of the four figures, a strongazimuthal dependence is evident.

The diagonal component Vdiag is computed for these measurements and isshown in FIGS. 8A-8D. As before, FIGS. 8A and 8B show the real andimaginary parts of the response signal for the parallel pair, whileFIGS. 8C and 8D show the real and imaginary parts of the response signalfor the perpendicular pair. Also shown as dashed lines are thesinusoidal curves that have been fit to the diagonal components, fromwhich parameters φ (with some ambiguity), A, and C can be determined.

Similarly, the cross component Vcc is computed for these measurementsand is shown in FIGS. 9A-9D. Real and imaginary parts for parallel andperpendicular antenna pairs are shown as before. The dashed linesrepresent the sinusoidal curves that best fit these components, therebyproviding parameter B and resolving the ambiguity of φ. From thediagonal component, the plane for the rotated x-axis can be determined.With an arbitrary choice for the x-axis direction in this plane, thecross component can be used to determine whether or not the x-axisdirection should be reversed. One way in which the ambiguity can beresolved is by comparing the imaginary and real parts of B as determinedby the parallel transmit-receive antenna pair. Specifically, if theimaginary part of B is greater than the real part, the x-axis directionshould be reversed.

Note that the magnitude of the cross-component signal is substantiallysmaller than the diagonal component. For this reason, the diagonalcomponent is preferred as the basis for estimating a casing distance.Specifically, the parameter A or the ratio of A/C may be used. FIGS.10A-10B are graphs of the logarithm of A/C versus casing distance forthe parallel and perpendicular transmit-receive antenna pairs,respectively. FIGS. 11A-11B are graphs of the logarithm of A versuscasing distance for the parallel and perpendicular transmit receiveantenna pairs, respectively. A clear dependence is evident, enabling astraightforward estimation of casing distance from the toolmeasurements.

It is expected that the system range and performance can be extendedwith the use of multiple receive antenna stations and/or multipletransmit antenna stations. In many situations, it may not be necessaryto perform explicit distance and direction calculations. For example,the signal components may be extracted and converted to pixel colors orintensities and displayed as a function of tool position and azimuth.Assuming the casing string is within detection range, it will appear asa bright (or, if preferred, a dark) band in the image. The color orbrightness of the band indicates the distance to the casing string, andthe position of the band indicates the direction to the casing string.Thus, by viewing such an image, a driller can determine in a veryintuitive manner whether the new borehole is drifting from the desiredcourse and he or she can quickly initiate corrective action. Forexample, if the band becomes dimmer, the driller can steer towards thecasing string. Conversely, if the band increases in brightness, thedriller can steer away from the casing string. If the band deviates fromits desired position directly above or below the casing string, thedriller can steer laterally to re-establish the desired directionalrelationship between the boreholes.

Numerous other variations and modifications will become apparent tothose skilled in the art once the above disclosure is fully appreciated.For example, the foregoing discussion has focused on a logging whiledrilling implementation, but the disclosed techniques would also besuitable for wireline tool implementation. The discussion provides forrotation of the tool (and its antennas), but multi-component antennameasurements can be used to obtain virtually-steered antennameasurements without requiring rotation of the tool or antennas. It isintended that the following claims be interpreted to embrace all suchvariations and modifications.

1. A method for detecting a conductive feature from within a borehole,the method comprising: transmitting an electromagnetic signal from afirst antenna on a downhole logging tool; measuring a response signalwith a second antenna on the downhole logging tool; repeating saidtransmitting and measuring to obtain an azimuthal dependence of saidresponse signal; determining a diagonal component of said azimuthaldependence; and using said diagonal component to estimate a distance toa casing string.
 2. The method of claim 1, further comprisingdetermining a casing string direction from the downhole logging tool. 3.The method of claim 2, wherein said determining a casing stringdirection includes fitting a sinusoid to the diagonal component.
 4. Themethod of claim 3, wherein said determining a casing string directionfurther includes determining a cross component of said azimuthaldependence.
 5. The method of claim 4, wherein said determining a casingstring direction still further includes fitting a sinusoid to the crosscomponent.
 6. The method of claim 5, wherein said sinusoid fit to thecross component has a complex amplitude, and wherein said determining acasing string direction includes comparing a real part of said amplitudeto a complex part of said amplitude.
 7. The method of claim 1, whereinthe diagonal component at bin β_(i) is proportional to${V_{R}^{T}( \beta_{i} )} + {V_{R}^{T}( \beta_{i + \frac{N}{2\;}} )}$where V_(R) ^(T) (β_(i)) represents an average signal measurementassociated with an azimuthal bin β_(i) and V_(R) ^(T)(β_(i±N/2))represents a signal measurement associated with a bin 180° away fromazimuthal bin β_(i).
 8. The method of claim 4, wherein the crosscomponent at bin β_(i) is proportional to${V_{R}^{T}( \beta_{i} )} - {V_{R}^{T}( \beta_{i \pm \frac{N}{2}} )}$where V_(R) ^(T)(β_(i)) represents an average signal measurementassociated with an azimuthal bin β_(i) and V_(R) ^(T)(β_(i±N/2))represents a signal measurement associated with a bin 180° away fromazimuthal bin β_(i).
 9. The method of claim 1, wherein the first andsecond antennas are each tilted relative to a tool axis.
 10. The methodof claim 9, wherein the first and second antennas are parallel.
 11. Themethod of claim 1, wherein the first and second antennas areperpendicular.
 12. A downhole ranging tool that comprises: a rotationalposition sensor; at least one transmit antenna to transmitelectromagnetic signals into a surrounding formation; at least onereceive antenna to receive response signals from the surroundingformation; and at least one processor that: determines average responsesignals for each of multiple rotational positions; extracts a diagonalcomponent from said average response signals; and estimates a distanceto a casing string based at least in part on said diagonal component.13. The tool of claim 12, wherein the at least one processor furtherfinds a direction to the casing string based at least in part on saiddiagonal component.
 14. The tool of claim 13, wherein as part of findingthe direction to the casing string, the at least one processor extractsa cross component from said average response signals.
 15. The tool ofclaim 14, wherein as part of finding the direction, the at least oneprocessor fits a complex sinusoid to the cross component and compares areal part of the complex sinusoid's amplitude to an imaginary part. 16.The tool of claim 12, wherein the diagonal component at bin β_(i) isproportional to${V_{R}^{T}( \beta_{i} )} + {V_{R}^{T}( \beta_{i \pm \; \frac{N}{2}} )}$where V_(R) ^(T)(β_(i)) represents an average signal measurementassociated with an azimuthal bin β_(i) and V_(R) ^(T)(β_(i±N/2))represents a signal measurement associated with a bin 180° away fromazimuthal bin β_(i).
 17. The tool of claim 14, wherein the crosscomponent at bin β_(i) is proportional to${V_{R}^{T}( \beta_{i} )} - {V_{R}^{T}( \beta_{i \pm \frac{N}{2}} )}$where V_(R) ^(T)(β_(i)) represents an average signal measurementassociated with an azimuthal bin β_(i) and V_(R) ^(T)(β_(i±N/2))represents a signal measurement associated with a bin 180° away fromazimuthal bin β_(i).
 18. The tool of claim 12, wherein at least one ofthe transmit and receive antennas is tilted relative to a tool axis. 19.The tool of claim 18, wherein the transmit and receive antennas areparallel.
 20. The tool of claim 18, wherein the transmit and receiveantennas are perpendicular.